DIGITAL IMAGE DIMENSION AND SPACE REDUCTION WITH CONTRACTION MAPPING

A digital image can be represented as two dimensional arrays of pixels. In this paper a simple but representative system of contractive mapping on Euclidean plane, called the digital plane is used for image processing to generate images with reduced dimension occupying less storage space and can be efficiently transmitted. With suitable matrix metric and contraction mapping, size of the original image matrix is diminished significantly by reducing the order of sub matrices and hence images of reduced size without much compromise to the quality of image is obtained. The variations between the original and contracted image are not too pronounced when the images are of large size and are seen on small screen (mobile, tablets etc.)


INTRODUCTION
Contraction is a mapping which reduces the distance (function). In 1922 an important theorem in metric space theory called Contraction Mapping Principle was given by Banach [1]. The concept is useful in the existence and uniqueness theory and considerably forms the foundation of range of image processing tools. The properties of digital images are characterized with tools from 3638 NEELAM YADAV, PIYUSH KUMAR TRIPATHI, SAJJAN LAL MAURYA algebraic topology. The field of digital topology was founded by Rosenfeld [2] and contributed fundamentally to variety of applications like image processing, pattern recognition, developed the notion of digital continuity for 2D and 3D digital images. The topological properties of digital images are studied as discretized arrays of two or more dimensions [3]. A digital metric space is complete [4]. Boxer [5] gave the digital versions of several notions from topology and studied a variety of digital continuous functions. Lefschetz fixed point theorem for digital images was proved by Ege and Karaca [6,7], also showed that sphere-like digital images have the fixed point property. Han [8] refined and improved various notions and assertions given in [6]. With advanced technology in digital cameras, digital imaging system has proliferated in the past few years. The demand of storing huge amount of large sized image files, the call for space to store them is increasing. Various lossless and lossy compression techniques are available [9] which facilitates in reducing the size of image data files. The authors put forward contraction method to reduce the size of image files as an application to contractive mappings. By reducing the total number of pixels, image resizing (reduction) can be achieved. An input image is of size × and objective is to obtain an image of proportionately reduced size. The original digital image is processed to reduce its size and the image when finally displayed on the screen is not exactly same as the original because it is subjected to distortions in trade off the size reduction, but will be the good representative of the original image. The quality of the processed image is confirmed not only by subjective method which involve human being to evaluate the quality of the image but also by the objective image quality metrics which numerically calculates the quality measure as PSNR [10]. In this paper 8 bit test image is used for illustration, therefore = 255.

PROPOSED ALGORITHM FOR IMAGE CONTRACTION
Various partitioning techniques are available in literature [9,11]  Consider , the digital image space, be the contraction mapping defined on it along with distance function . Here the contraction mapping Principle is given by: The steps of the proposed algorithm is listed below which involves local operations on each square fixed width block which are performed in parallel on every element (pixel value) and to  (1) and corresponding contracted sub-matrix 1 of dimension 1 × 1 is obtained. Arrange each contracted sub matrix in the same sequence as that of the parent sub matrix, subsequently obtain contracted matrix B.
The above steps can be understood as

ILLUSTRATION
Python 3.7 on spyder IDE is used to implement the proposed algorithm. The image (Fig. 2) is used as an input to the algorithm. Input reference image I is converted to subsequent × matrix where each element correspond to the intensity of that pixel. Matrix is then partitioned into several blocks (sub-matrices) of fixed block size ∈ ( ≥ 2). This is image segmentation. Each block is worked on without the reference to the others, independently and if programmed can be processed concurrently. The various steps involved are illustrated below.
Consider a small 16 × 16 pixel region of the original image (Fig. 2). The first matrix is

RESULT AND ANALYSIS
The image (Fig. 2)  Using the pixels of reduced output image, the reconstructed images (Fig. 4a, 4b and 4c) are obtained to match the dimension of . Quality of reconstructed images is given by the PSNR (listed in table 1). But using the subjective quality measure it is clear from Fig. 4a, 4b and 4c that as the increases, the quality of reconstructed images minifies.
The properties of contracted images are summarized in Table 1. Space saved is calculated with reference to image I, repetitive application of the preset function can be implemented to further diminish the image size. The 114 × 85 size (Fig. 4c) can be achieved either by considering partitioning with block size 4 × 4 or by repeated application of the scheme (iteratively) twice by considering block size 2 × 2 in each step. In both the cases the contracted image will be approximately the same. The quality of processed digital image can be done subjectively and objectively. PSNR is one of the widely used objective quality measure [10].

CONCLUSION
The metric is defined suitably on a continuous digital space which is then discretized as pixels for image processing. The concept of Contraction mapping on composition of functions is defined and has been used to get contracted images. It is noticed the original image size contracted without much compromise in image quality. Thus, the obtained contracted image is of reduced size and utilizes less space for storage and therefore comparatively easier to transmit.
The scheme is implemented to various block size to examine the extent an image can be contracted. Also, the designed function is iterated repeatedly on input image, the obtained final contracted image is of considerable quality if the input image is of large size and/or is an image with less variation in colors.